Calculus question
3 months ago 5
CALCULUSIREVISONPAPER2.pdf
CALCULUSIREVISONPAPER2.pdf
UNIVERSITY EXAMINATION
INSTRUCTION ANSWER QUESTION ONE (COMPULSORY) AND ANY OTHER TWO QUESTIONS.
QUESTION ONE (30MARKS)
a) Evaluate the following limits
i)
+→ tt t
t sin 2lim
0 (2marks)
ii)
−−+ → x
xx x
11lim 0
(3marks)
b) Use first principle of differentiation to find the rate of change of y(x) with respect to x
given. ( ) x
xxy 31+
= (5marks)
c) For a firm under perfect competition it is given that P=sh. 19,
27285 3
)( 2 3
+++= qqqqC where P represents price per unit, q units of output and
𝐶𝐶(𝑞𝑞) total cost. Find i) Quantity produced at which profit will be maximum ii) The amount of maximum profit (7marks) d) Show that 24 2)( xxxf −= , ]2,2[− satisfies the hypothesis of Rolle’s theorem, find
all x in the interval that satisfy the conclusion of the theorem (4marks)
e) Find dx dy given
i) ( )x x y 1cos−= (3marks)
ii) 643 2 +−= ttx and 416 ty −= (3marks)
iii) xxy sin= (3marks)
QUESTION TWO (20MARKS)
a) identify and classify the extreme values of the function ( )xxxxf 3632)( 23 −−= (6marks)
b) Determine the values of a and b for which the function g(x) is continuous everywhere on real number line
𝑔𝑔(𝑥𝑥) = � 5 𝑖𝑖𝑖𝑖 𝑥𝑥 ≤ 3
𝑎𝑎𝑥𝑥 − 𝑏𝑏 𝑖𝑖𝑖𝑖 3 < 𝑥𝑥 < 4 11 𝑖𝑖𝑖𝑖 5 ≥ 𝑥𝑥
(6marks)
c) Given that )( )()(
xv xuxf = , ( ) 0≠xv , show that ( ) ( ) ( ) ( )
( )( )2
')(' xv
xvxuxvxu dx
xdf − = hence find
dx dy given
x xy
cos1 sin +
= (8marks)
QUESTION THREE (20MARKS)
a) Find the equation of the tangent to the curve 132 22 =++ yxyx through the point (2,1) (3marks)
b) Investigate whether [ ]5,2563)( 2 −−+= xxxf satisfies the mean value theorem (5marks)
c) Find the rate of change of y(x) with respect to x given i) ( )323 102ln xyxxyx =+ (4marks)
ii) 231 tx += 2
2
31 2
t ty
+ −
= (5marks)
iii) )2tan(ln63 3
xexy x −= + (3marks)
QUESTION FOUR(20MARKS)
a) Evaluate ( )xf x 2 lim →
given 𝑖𝑖(𝑥𝑥) = � 𝑥𝑥 2 − 4𝑥𝑥 + 6, 𝑥𝑥 < 2
−𝑥𝑥2 + 4𝑥𝑥 − 2, 𝑥𝑥 ≥ 2 (4marks)
b) Discuss the continuity of the function 𝑖𝑖(𝑥𝑥) = � 1 23 2
−
−−
x xx
5 𝑥𝑥 = 1
𝑥𝑥 ≠ 1
At 𝑥𝑥 = 1 (5marks) c) The ordering and transportation cost C of the components used in the manufacturing a
certain product is given by
+ +=
30 200100 2 x
x x
C 1 ≤ 𝑥𝑥 where C is measured in
thousands of dollars and x is the order size in hundreds. Find the order size that minimizes the cost (6marks)
d) Gravel is falling in a conical pile at the rate of 100 cubic feet per minute. Find the rate of change of height of the pile when the height is 10 feet. (assume that the coarseness of the gravel is such that the radius of the cone is equal to its height) (5marks)
CALCULUSIREVISONPAPER2.pdf
UNIVERSITY EXAMINATION
INSTRUCTION ANSWER QUESTION ONE (COMPULSORY) AND ANY OTHER TWO QUESTIONS.
QUESTION ONE (30MARKS)
a) Evaluate the following limits
i)
+→ tt t
t sin 2lim
0 (2marks)
ii)
−−+ → x
xx x
11lim 0
(3marks)
b) Use first principle of differentiation to find the rate of change of y(x) with respect to x
given. ( ) x
xxy 31+
= (5marks)
c) For a firm under perfect competition it is given that P=sh. 19,
27285 3
)( 2 3
+++= qqqqC where P represents price per unit, q units of output and
𝐶𝐶(𝑞𝑞) total cost. Find i) Quantity produced at which profit will be maximum ii) The amount of maximum profit (7marks) d) Show that 24 2)( xxxf −= , ]2,2[− satisfies the hypothesis of Rolle’s theorem, find
all x in the interval that satisfy the conclusion of the theorem (4marks)
e) Find dx dy given
i) ( )x x y 1cos−= (3marks)
ii) 643 2 +−= ttx and 416 ty −= (3marks)
iii) xxy sin= (3marks)
QUESTION TWO (20MARKS)
a) identify and classify the extreme values of the function ( )xxxxf 3632)( 23 −−= (6marks)
b) Determine the values of a and b for which the function g(x) is continuous everywhere on real number line
𝑔𝑔(𝑥𝑥) = � 5 𝑖𝑖𝑖𝑖 𝑥𝑥 ≤ 3
𝑎𝑎𝑥𝑥 − 𝑏𝑏 𝑖𝑖𝑖𝑖 3 < 𝑥𝑥 < 4 11 𝑖𝑖𝑖𝑖 5 ≥ 𝑥𝑥
(6marks)
c) Given that )( )()(
xv xuxf = , ( ) 0≠xv , show that ( ) ( ) ( ) ( )
( )( )2
')(' xv
xvxuxvxu dx
xdf − = hence find
dx dy given
x xy
cos1 sin +
= (8marks)
QUESTION THREE (20MARKS)
a) Find the equation of the tangent to the curve 132 22 =++ yxyx through the point (2,1) (3marks)
b) Investigate whether [ ]5,2563)( 2 −−+= xxxf satisfies the mean value theorem (5marks)
c) Find the rate of change of y(x) with respect to x given i) ( )323 102ln xyxxyx =+ (4marks)
ii) 231 tx += 2
2
31 2
t ty
+ −
= (5marks)
iii) )2tan(ln63 3
xexy x −= + (3marks)
QUESTION FOUR(20MARKS)
a) Evaluate ( )xf x 2 lim →
given 𝑖𝑖(𝑥𝑥) = � 𝑥𝑥 2 − 4𝑥𝑥 + 6, 𝑥𝑥 < 2
−𝑥𝑥2 + 4𝑥𝑥 − 2, 𝑥𝑥 ≥ 2 (4marks)
b) Discuss the continuity of the function 𝑖𝑖(𝑥𝑥) = � 1 23 2
−
−−
x xx
5 𝑥𝑥 = 1
𝑥𝑥 ≠ 1
At 𝑥𝑥 = 1 (5marks) c) The ordering and transportation cost C of the components used in the manufacturing a
certain product is given by
+ +=
30 200100 2 x
x x
C 1 ≤ 𝑥𝑥 where C is measured in
thousands of dollars and x is the order size in hundreds. Find the order size that minimizes the cost (6marks)
d) Gravel is falling in a conical pile at the rate of 100 cubic feet per minute. Find the rate of change of height of the pile when the height is 10 feet. (assume that the coarseness of the gravel is such that the radius of the cone is equal to its height) (5marks)